2 SUBSTITUTION OF POWER SERIES IN A POWER SERIES 201

(9.1.6) Suppose two power series (0vzv) and (bvzv) are absolutely summable
and have the same sum in a poly disk P; then av = bvfor every v e W.

Use induction on p; for p = 1, the result follows at once from (9.1.5).
Taking the difference of the two power series, we can assume bv = 0 for

every v; applying (9.1.4) with q=p-\, we have £ #n(z')z£ = 0, hence

gn(z') = 0 for every n and every z' in the projection F of P on Kp-1; the
induction hypothesis applied to each gn yields then av = 0 for every v.

PROBLEMS

1. Let (cvzv) be a power series in p variables z, (1 ^i^p); let a = (als ...,0P)EKP.
In order that a real number r > 0 be such that, for any / e K such that |/| < r, the series
(cv(tat)ni - • - (tap)np) be absolutely summable, it is necessary and sufficient that

1 ,£

log r + n logllcvll -f £ nt log|fl, ^ 0

for all but a finite number of indices v = (n^ . . . , np) (apply (9.1.2).)

In particular, for/? = 1, there is a largest number R 5= 0 (the "convergence radius,"
which may be +00) such that the series (cnzn) is convergent for z\ <R, and that
number is given by 1/R = lim (sup(||cn+k||1/<n+fc))), which is also written lim • sup||cn||1/fl

n-Kğ ft>0 JI-+00

(cf. Section 12-7). When in particular lim ||cn||1/fl exists, it is equal to 1/R.

n-*oo

2. Give examples of power series in one complex variable, having a radius of convergence
R = 1 (Problem 1) and such that:

(1) the series is normally convergent for |z| = R;

(2) the series is convergent for some z such that |z = R, but not for other points of
that circle;

(3) the series is not convergent at any point of |z| = R.

3. Give an example of a power series in two variables, which is absolutely summable at

— — t — — I . (Replace z by

ziZ2 in a power series in one variable.)

4. Let (CflZ"), (dnzn) be two power series in one variable with scalar coefficients; if their
radii of convergence (Problem 1 ) are R and R', and neither R nor R' is 0, then the
radius of convergence R" of the power series (cndnzn) is at least RRX (taken equal to
-f oo if R or Rx is + oo). Give an example in which R" > RRX.

2. SUBSTITUTION OF POWER SERIES IN A POWER SERIES

Let Q be a polydisk of center 0 in K4, and suppose the p power series
in q variables (b^u*) with scalar coefficients are absolutely summable
in Q (with ju = (ml9 ..., mq)9 u = (ui9..., uq), uğ =u^ ... u%<). We writepelled to attribute for z ^ 0, two distinct